Theorem igenval2 33865

Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
igenval2.1	|- G = ( 1st ` R )
igenval2.2	|- X = ran G

Assertion

Ref	Expression
igenval2	|- ( ( R e. RingOps /\ S C_ X ) -> ( ( R IdlGen S ) = I <-> ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) )

Distinct variable groups:   R, j   S, j   j, I

Allowed substitution hints:   G( j)   X( j)

Proof of Theorem igenval2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval2.1	. . . . 5 |- G = ( 1st ` R )
2		igenval2.2	. . . . 5 |- X = ran G
3	1, 2	igenidl 33862	. . . 4 |- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) e. ( Idl ` R ) )
4	1, 2	igenss 33861	. . . 4 |- ( ( R e. RingOps /\ S C_ X ) -> S C_ ( R IdlGen S ) )
5		igenmin 33863	. . . . . . 7 |- ( ( R e. RingOps /\ j e. ( Idl ` R ) /\ S C_ j ) -> ( R IdlGen S ) C_ j )
6	5	3expia 1267	. . . . . 6 |- ( ( R e. RingOps /\ j e. ( Idl ` R ) ) -> ( S C_ j -> ( R IdlGen S ) C_ j ) )
7	6	ralrimiva 2966	. . . . 5 |- ( R e. RingOps -> A. j e. ( Idl ` R ) ( S C_ j -> ( R IdlGen S ) C_ j ) )
8	7	adantr 481	. . . 4 |- ( ( R e. RingOps /\ S C_ X ) -> A. j e. ( Idl ` R ) ( S C_ j -> ( R IdlGen S ) C_ j ) )
9	3, 4, 8	3jca 1242	. . 3 |- ( ( R e. RingOps /\ S C_ X ) -> ( ( R IdlGen S ) e. ( Idl ` R ) /\ S C_ ( R IdlGen S ) /\ A. j e. ( Idl ` R ) ( S C_ j -> ( R IdlGen S ) C_ j ) ) )
10		eleq1 2689	. . . 4 |- ( ( R IdlGen S ) = I -> ( ( R IdlGen S ) e. ( Idl ` R ) <-> I e. ( Idl ` R ) ) )
11		sseq2 3627	. . . 4 |- ( ( R IdlGen S ) = I -> ( S C_ ( R IdlGen S ) <-> S C_ I ) )
12		sseq1 3626	. . . . . 6 |- ( ( R IdlGen S ) = I -> ( ( R IdlGen S ) C_ j <-> I C_ j ) )
13	12	imbi2d 330	. . . . 5 |- ( ( R IdlGen S ) = I -> ( ( S C_ j -> ( R IdlGen S ) C_ j ) <-> ( S C_ j -> I C_ j ) ) )
14	13	ralbidv 2986	. . . 4 |- ( ( R IdlGen S ) = I -> ( A. j e. ( Idl ` R ) ( S C_ j -> ( R IdlGen S ) C_ j ) <-> A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) )
15	10, 11, 14	3anbi123d 1399	. . 3 |- ( ( R IdlGen S ) = I -> ( ( ( R IdlGen S ) e. ( Idl ` R ) /\ S C_ ( R IdlGen S ) /\ A. j e. ( Idl ` R ) ( S C_ j -> ( R IdlGen S ) C_ j ) ) <-> ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) )
16	9, 15	syl5ibcom 235	. 2 |- ( ( R e. RingOps /\ S C_ X ) -> ( ( R IdlGen S ) = I -> ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) )
17		igenmin 33863	. . . . . 6 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) /\ S C_ I ) -> ( R IdlGen S ) C_ I )
18	17	3adant3r3 1276	. . . . 5 |- ( ( R e. RingOps /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> ( R IdlGen S ) C_ I )
19	18	adantlr 751	. . . 4 |- ( ( ( R e. RingOps /\ S C_ X ) /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> ( R IdlGen S ) C_ I )
20		ssint 4493	. . . . . . . 8 |- ( I C_ |^| { i e. ( Idl ` R ) | S C_ i } <-> A. j e. { i e. ( Idl ` R ) | S C_ i } I C_ j )
21		sseq2 3627	. . . . . . . . 9 |- ( i = j -> ( S C_ i <-> S C_ j ) )
22	21	ralrab 3368	. . . . . . . 8 |- ( A. j e. { i e. ( Idl ` R ) | S C_ i } I C_ j <-> A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) )
23	20, 22	sylbbr 226	. . . . . . 7 |- ( A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) -> I C_ |^| { i e. ( Idl ` R ) | S C_ i } )
24	23	3ad2ant3 1084	. . . . . 6 |- ( ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) -> I C_ |^| { i e. ( Idl ` R ) | S C_ i } )
25	24	adantl 482	. . . . 5 |- ( ( ( R e. RingOps /\ S C_ X ) /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> I C_ |^| { i e. ( Idl ` R ) | S C_ i } )
26	1, 2	igenval 33860	. . . . . 6 |- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) = |^| { i e. ( Idl ` R ) | S C_ i } )
27	26	adantr 481	. . . . 5 |- ( ( ( R e. RingOps /\ S C_ X ) /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> ( R IdlGen S ) = |^| { i e. ( Idl ` R ) | S C_ i } )
28	25, 27	sseqtr4d 3642	. . . 4 |- ( ( ( R e. RingOps /\ S C_ X ) /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> I C_ ( R IdlGen S ) )
29	19, 28	eqssd 3620	. . 3 |- ( ( ( R e. RingOps /\ S C_ X ) /\ ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) -> ( R IdlGen S ) = I )
30	29	ex 450	. 2 |- ( ( R e. RingOps /\ S C_ X ) -> ( ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) -> ( R IdlGen S ) = I ) )
31	16, 30	impbid 202	1 |- ( ( R e. RingOps /\ S C_ X ) -> ( ( R IdlGen S ) = I <-> ( I e. ( Idl ` R ) /\ S C_ I /\ A. j e. ( Idl ` R ) ( S C_ j -> I C_ j ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   |^|cint 4475   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   RingOpscrngo 33693   Idlcidl 33806   IdlGen cigen 33858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-rngo 33694  df-idl 33809  df-igen 33859

This theorem is referenced by:  prnc  33866